Sounds like you may need a second opinion.
Trouble is - you cannot pull the code, look it up and see what needs to be changed - there are diagnostic steps that must be followed or you'll end up just wasting money swapping parts.
A P0420 code is described as - Catalyst System Efficiency Below Threshold (Bank 1). What this means is that the oxygen sensor located after the catalytic converter is detecting that the converter is not working as efficiently as it should be. This would indicate several possible causes:
- O2 sensor is dead or lazy in response
- engine coolant temperature sensor is faulty or has a bad connection
- possible physical damage to exhaust manifold, catalytic converter, or the exhaust pipe
- ignition timing is off (retarded spark timing)
- and a few more
The MAF sensor (your model has MAF "Mass Air Flow" not MAP "Manifold Absolute Pressure" - two completely different systems) could be the cause - but it seems you have a more complicated issue here or that the real problem is masked by these common ones.
Not sure what you meant by "Do i pay for the parts put in by this mechanic?" - if you meant that you may have paid for parts that were not necessary or fixed the problem - hard to say for sure without looking at the car, but I'd say NO given that the same OBD-II codes keep coming back. Hopefully you got the original parts back - they can be tested later to see if they are actually bad or not. None of this will be covered by the factory warranty unless you have additional coverage that states it will be covered - except for possibly the catalytic converter (Federal Emissions coverage protects most parts for 36months/36K miles and "specified major emission control components" for 8years/80K miles).
Good Luck.